Analog electrical circuit theory provides mathematical relationships between voltage v, current i, charge q, and magnetic flux φ.
There are six unordered pairs that may be selected from this set of four variables. The well-known electrical properties of the most familiar passive elements (namely resistors, inductors, and capacitors) provide relationships between three of these six pairs of variables.
For example, an ideal resistor with resistance R is a two-terminal passive circuit element defined by the relationship between voltage v(t) and current i(t):dv=Rdi. 
An ideal inductor with inductance L is a two-terminal passive circuit element defined by the relationship between flux φ(t) and current i(t):dφ=Ldi. 
An ideal capacitor with capacitance C is a two-terminal passive circuit element defined by the relationship between charge q(t) and voltage v(t):dq=Cdv. 
A fourth of the six possible relationships is provided by the definition of charge q(t) (as the time integral of current):dq=idt. 
In addition, a fifth relationship is provided by the definition of flux φ(t) (as the time integral of voltage):dφ=vdt. 
In a 1971 paper entitled “The Missing Circuit Element” (Chua 1971), Leon Chua of the University of California at Berkeley hypothesized the existence of a fourth two-terminal passive circuit element defined by the relationship between flux φ(t) and charge q(t):dφ=M(q)dq. 
Chua demonstrated that no combination of the three most familiar passive electrical elements (resistors, inductors, and capacitors) could duplicate the function of this hypothesized fourth passive circuit element. Chua gave the name “memristor” to this circuit element. A memristor's memristance, M(q), is a function of charge Like resistance, memristance is measured in Ohms.
Because no physical realization of Chua's hypothesized circuit element as a single passive component existed in 1971, experiments involving the behavior of memristors were conducted, for a number of years, by emulating the memristor by rather complex and impractical arrangements of active elements (using, in one instance, 15 transistors along with additional passive circuit elements).
In a 2008 article in Nature entitled “The Missing Memristor Found,” (hereinafter “Strukov Article”) Dmitri B. Strukov, Gregory S. Snider, Duncan R. Stewart, and R. Stanley Williams of HP Labs announced the development of a physical prototype of a memristor as a passive component. HP Labs' memristor is a nanoscale device with layers of titanium-dioxide and platinum.
The characteristics of HP Labs' memristor are described as follows. First, memristors are passive devices. Second, memristance is a continuous analog quantity that changes as a function of the time integral of the current that has passed through the memristor. Third, the memristor retains the analog value of this time integral even when no current is flowing. Indeed, the name “memristor” (short for “memory resistor”) reflects the fact that the memristor is a non-volatile memory device. Thus, memristors may be used to store analog information. Fourth, when a certain amount of current has passed through the memristor in a particular direction, the memristor ceases to further integrate current in that direction. Thus, memristors have a maximum-resistance “off” state and a minimum-resistance “on” state (permitting them to be used to store binary digital information). In the memristor developed by HP Labs, the time integration of current ceases because dopants can no longer move. Fifth, the memristor's continuous analog behavior resembles that of neural synapses and other processes known to exist in nature. Sixth, memristors are two-terminal devices.
FIG. 1 shows a schematic of a single memristor M 100. In the present disclosure, we adopt the convention of using the term “positive terminal” to refer to the terminal 110 which, when a positive current flows through the memristor in the direction from the terminal designated as the positive terminal to the memristor's other terminal (i.e., its negative terminal 120), the doped, low resistance portion 250 of FIG. 2 increases and thus memristance moves toward its low-end value of RON.
FIG. 2 illustrates the memristor developed by HP Labs. A semiconductor film of thickness D 230 is sandwiched between platinum contact 201 (associated with terminal 200) and platinum contact 211 (associated with terminal 210). The memristor's memristance depends on the concentration of dopants (e.g., positive ions) between the two metal contacts. The degree of concentration of dopants in the semiconductor film between the two metal contacts is represented in FIG. 2 as a geographic boundary 270 dividing the semiconductor film into two portions. The memristor's total memristance depends on the location of the boundary 270. The memristor shown in FIG. 2 is in a partially doped state. In particular, the semiconductor film has a first portion 250 (shown to the left side of boundary 270) having a high concentration of dopants and, therefore, low resistance. The semiconductor film has a second portion 260 (to the right of boundary 270) having a low (essentially zero) dopant concentration and appreciably higher resistance. The quantities w 220 and D−w 280 indicate the location of the boundary in FIG. 2. The fraction of the memristor that is doped is w/D, and the fraction of the memristor that is undoped is [D−w]/D. If boundary 270 is at the far left of FIG. 1 (i.e., the device is in the totally undoped state), the resistance of the device is at its maximal value ROFF. If boundary 270 is at the far right, the resistance of the device is at its minimal value RON. In practice, RON<<ROFF. The memristance of the device, M(w), at a particular time t depends on the changing value of w(t), and is as follows:
      M    ⁡          (              w        ⁡                  (          t          )                    )        =                    R        ON            ⁢                        w          ⁡                      (            t            )                          D              +                            R          OFF                ⁡                  (                      1            -                                          w                ⁡                                  (                  t                  )                                            D                                )                    .      
A difference in voltage v(t) between the voltage at terminal 200 and the voltage at terminal 210 causes a current to flow through the memristor. The current flow, in turn, causes dopants to drift. The dopant drift changes w(t) 220, thereby adjusting the boundary 270 in FIG. 2. As explained in the Strukov Article, for the case of ohmic electronic conductance and linear ionic drift in a uniform field with average ion mobility uv,
            v      ⁡              (        t        )              =                  i        ⁡                  (          t          )                    ⁡              [                                            R              ON                        ⁢                                          w                ⁡                                  (                  t                  )                                            D                                +                                    R              OFF                        ⁡                          (                              1                -                                                      w                    ⁡                                          (                      t                      )                                                        D                                            )                                      ]              and                    ⅆ                  w          ⁡                      (            t            )                                      ⅆ        t              =                  u        v            ⁢              i        ⁡                  (          t          )                    ⁢                        R          ON                /                  D          .                    
Integrating both sides of the previous equation yieldsw(t)=uvq(t)RON/D. M(q)=ROFF[1−q(t)uvRON/D2].
Based on the above characteristics, memristors have been predicted to find fruitful applications in areas such as ultra-dense non-volatile memory cells, crossbar memory, and logic. Because of the resemblance of the memristor's continuous analog behavior to neural synapses, it has also been predicted that memristors may find applications in the field of analog memories, pattern recognition, and artificial intelligence. Because the memristor is a two-terminal passive device, circuit layout may be made more efficient in certain situations. Because the memristor is a non-volatile memory device, memristors may find many applications where energy usage is important.
Substituting the above equation for w(t) into the above equation for v(t) yields the memristance of the system which, when RON<<ROFF, simplifies to